In an earlier paper the authors considered r-almost s-uniform trees, i.e. rooted planar trees T such that the root has r successors, and every other vertex has s suc- cessors. They considered binary functions f : V (T ) → {−1, +1} defined on the set V (T ) of the vertices of such a tree T and studied the pseudorandomness of binary functions of this type. Here the authors extend the problem to general rooted plane trees: the measures of pseudorandomness of binary functions defined on trees of this type are introduced; the connection between these measures is analyzed; the size of these measures for truly random binary functions is studied; binary functions with strong pseudorandom properties are constructed; pseudorandom properties of impor- tant special binary functions are studied.